It is reasonable to assume that this is a specification for a number: after all, there are a finite number of sentences of less than eleven words, and some finite subset of them specify unique positive integers, so there is clearly some positive number that is the smallest integer not in that finite set.

This is clearly paradoxical, and seems to indicate that "nameable in under ten words" is not cleanly enough defined.

Berry had provided the original idea in a letter to Russell about the less specific "the first ordinal that cannot be named in a finite number of words".

Berry'sparadox involves describing something using a description whose form is in apparent contradiction with the meaning of the description.

Berry'sparadox is an example of a family of supposed paradoxes which have in fact no paradoxical force and require no therapy.

In the case of Berry'sparadox, the partial function is the function which is supposed to assign (positive) integers to some domain of objects we call "phrases".

www.cs.yorku.ca /~peter/Berry.html (1030 words)

From Frege To Godel: von Heijenoort(Site not responding. Last check: 2007-10-20)

The Burali-Forti paradox deals with the "greatest ordinal"--which is obtained by assuming the set of ordinals is well-ordered [and, of course, that it is a set!]--which must be a member of the set of ordinals and simultaneously greater than any ordinal in the set.

Peano (1906a) rejected Richard's paradox as a paradox of linguistics, not mathematics.

The argument is based on the Liar paradox, except instead of considering a statement expressing "I am not true," he considers a statement expressing "I am not provable." The diagonalization used to construct such a statement is reminiscent of Cantor's diagonal procedure and Richard's paradox.

This version of the paradox applies only to the natural numbers, as it depends on mathematical induction, which is only applicable to sets that are well-ordered; the argument does not apply to the real numbers.

The Berryparadox is closely related, arising from a similar self-referential definition.

As the paradox lies in the definition of "interesting", it applies only to persons of sufficiently sophisticated taste in numbers: if one's view is that all numbers are boring, and one is unmoved by the observation that 0 is the smallest boring number, there's no paradox.

Russell's Paradox signaled the end of the period of youthful innocence in Cantor's Set Theory, during which it was believed that to every property could be associated a well-defined set, namely the set of all objects having that property.

Many people mistakenly believe that paradoxes such as Russell's are a consequence of self-reference, and that their message is that we should banish self-reference as inescapably paradoxical.

Berry'sParadox is another puzzle that seems to hinge on self-reference.

www.maa.org /devlin/devlin_11_03.html (1494 words)

Some paradoxes(Site not responding. Last check: 2007-10-20)

Eubulides, the Megarian sixth century B.C. Greek philosopher, and successor to Euclid, invented the paradox of the liar.

In this paradox, Epimenides, the Cretan, says, "All Cretans are liars." If he is telling the truth he is lying; and if he is lying, he is telling the truth.

The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless.

www.wordsmith.demon.co.uk /paradoxes/index.htm (3977 words)

Math Lair - Paradoxes(Site not responding. Last check: 2007-10-20)

The paradoxes listed below and most other mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity, or are a result of self-reference.

The paradox lies in the fact that there are valid reasons for choosing either (a) or (b).

The paradox lies in the fact that it doesn't make sense to have an infinite expectation, since it is only possible to win finite amounts of money.

A paradox, devised by G. Berry of the Bodleian Library at Oxford University in 1906, that involves statements of the form: "The smallest number not nameable in under ten words." At first sight, there doesn't seem anything particularly mysterious about this sentence.

Berry'sparadox shows that the concept of nameability is inherently ambiguous and a dangerous concept to be used without qualification.

A similar air of the paradoxical swirls around the notion of interesting numbers.

www.daviddarling.info /encyclopedia/B/Berrys_paradox.html (212 words)

Moving the Dark to Wholeness: The Elegies of Wendell Berry(Site not responding. Last check: 2007-10-20)

Berry's elegies are vital to his work because they provide focus for his typical themes, such as his recurring metaphors that link past and future generations through their common working of the land.

Berry tells us that Owen Flood's ``passion'' was ``to be true / to the condition of the Fall/ to live by the sweat of his face, to eat / his bread, assured that the cost was paid'' (237).

Berry ends his poem ``Rising'' with as deeply felt an ``apotheosis'' as we are likely to encounter in contemporary American poetry, and in it he sums up much of his attitude toward death and life.

Russell's paradox: the set of all sets that are not members of themselves.

(For the Berryparadox, this is not so obvious, but it is because it refers to the number of syllables in our language, and so it refers both to numbers and to the language we are using to talk about these numbers).

Berryparadox: "the least integer not nameable in fewer than nineteen syllables." In a type theory formal system, any property analogous to number of syllables will refer to lower types; it cannot be used to describe its own statement.

These ’paradoxes’ are misunderstandings intimately involved in the work of Gödel and others, when they are discussing some elements of modern ‘logic’ or establishing ‘proofs’.

But that is to be caught in the error of the ‘paradox’, or maybe this analysis is given expressly in order to remove your attention from the hand of the magician.

Russell’s ‘paradox’ is founded on the error of not distinguishing the formation of a new item, that is the forming of the catalogue, from the original items to be catalogued.

www.abelard.org /metalogic/metalogicA3.htm (8483 words)

[No title](Site not responding. Last check: 2007-10-20)

The paradox is that as soon as you find an integer, say m, that supposedly requires more characters to be specified than is in the sentence above you can turn around and say that the sentence above specifies it perfectly well.

Paradoxes of this kind usually point out problems with our reasoning, of some form, and so are occasionally the basis for a new proof or of a a reformulation of the accepted theory.

Berry'sparadox is the foundation of a proof by Greg Chaitin (and similar work by A. Kolmogorov and R. Solomonoff).

While such paradoxes may be resolved in time with better understanding, it is unlikely that the paradoxes mentioned here will be so easily resolved.

This paradox is quite representative of the general problem of the Social Dilemmas which I discuss here and has to do with the fact that an individual's vote has no significant impact on the outcome of an election.

The paradox is important in theory since some philosophers claim that Newcomb's Paradox and the Prisoner's Dilemma are essentially the same phenomenon.

Berry is also credited with the invention of the greeting card paradox--he would introduce himself with a card that on one side said:

Russell discussed each of these paradoxes (and several more) in his "Mathematical logic as based on the theory of types [Russell1908] (reprinted in [Heijenoort67]) and concludes that they do not affect the logical calculus which is incapable of expressing their character.

Again, these are semantical paradoxes unlike Russell's famous paradox of the set of all sets that do not contain themselves. This paradox is often recast as a question about a barber:

Zeno's Paradoxes are primarily about the possibility of motion, but more generally they are about the possibility of specifying the units, or atomic parts, of which either space or time, or indeed any continuum may be thought to be composed.

In fact an earlier paradox about the natural numbers had suggested that even they could not be consistently numbered: for they could be put into 1 to 1 correlation with the even numbers, for one thing, and yet there were surely more of them, since they included the odd numbers as well.

This paradox Cantor took to be avoided by his definition of the power of a set (N.B. not the power set of a set): his definition merely required two sets to be put into 1 to 1 correlation in order for them to have the same power.

www.iep.utm.edu /p/par-log.htm (8953 words)

Computing Papers on Paradox(Site not responding. Last check: 2007-10-20)

Paradoxes are centerpieces of the historical tension between philosophy and technique.

A Paradox was raised in [16] when a classical predator-prey model was used to solve the biological control problem.

However, in some situations in preference voting, where a vote is a summary of an individual s preferences, it is known that such voting scheme could outperform plurality voting when applied in the reconciliation of models.

The semantic paradoxes have been a trap for logicians who, in their attempts to solve the paradoxes, have tended to view the patterns of paradox as simpler and more predictable than they actually are.

This paradox can be used (in much the same way as Gödel [1931] used the Richard paradox, Tarski [1936] used the Liar paradox, and Chaitin [1966] used the Berryparadox) to motivate a limitative theorem.

This variation of Russell's paradox in terms of degrees of randomness suggests that infinite-valued generalizations of Rice's Theorem are worthy of further investigation.

The fact that we are wrongly told at the outset that such a barber exist just does not make it appear out of thin air in spite of the logical contradictions his very existence would imply.

This paradox was instrumental in clarifying the (Zermelo-Fraenkel) axioms of modern set theory: A "set" cannot be just anything we like.

Although this common sense approach is perfectly sound, its would seem that the basic distinction between finite and infinite sets ought not to depend explicitely on the properties of something as complicated as the integers.

The concept of an uninteresting number comes up in Berry'sParadox, which was, paradoxically, not invented by Berry, although G.G. Berry, an Oxford librarian, did suggest the idea to Bertrand Russell, who turned it into a serious topic for mathematical and philosophical inquiry and named it after his librarian friend.

Berry'sParadox is exemplified by the phrase, "The first natural number that cannot be named in fewer than fourteen words." That phrase contains 13 and purports to name a number that can't be named in fewer than 14 words, hence, the paradox.

Chaitin did to the BerryParadox what Goedel had famously done to the Liar Paradox in developing his results on the inherent incompleteness of arithmetic and other formal systems.

The Biology Club is a registered student organization open to all students interested in biology.

All students are eligible to enroll in the Biology Club with a minimal dues payment.

Berry College is home to the Tau Alpha chapter of Tri-Beta, the national biological honor society.

www.berry.edu /academics/science/biology/clubs.asp (620 words)

Berrys Paradox(Site not responding. Last check: 2007-10-20)

To clarify, the situation is a form of Berry'sParadox.

For the purposes of the original form of this paradox only (given below), definitions may be taken to be formed from words drawn from a finite dictionary (so that only finitely many definitions exist formed from a given number of syllables).

See http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html for a transscript of a lecture by GregoryChaitin on the subject of the Berryparadox (and how he tried to meet with KurtGoedel but never succeeded).